### Key Concepts

### 8.1 Simplify Expressions with Roots

**Square Root Notation**- $\sqrt{m}$ is read ‘the square root of
*m*’ - If
*n*^{2}=*m*, then $n=\sqrt{m},$ for $n\ge 0.$

- The square root of
*m*, $\sqrt{m},$ is a positive number whose square is*m*.

- $\sqrt{m}$ is read ‘the square root of
*n*^{th}Root of a Number- If ${b}^{n}=a,$ then
*b*is an*n*root of^{th}*a*. - The principal
*n*root of^{th}*a*is written $\sqrt[n]{a}.$ *n*is called the*index*of the radical.

- If ${b}^{n}=a,$ then
**Properties of $\sqrt[n]{a}$**- When
*n*is an even number and- $a\ge 0,$ then $\sqrt[n]{a}$ is a real number
- $a<0,$ then $\sqrt[n]{a}$ is not a real number

- When
*n*is an odd number, $\sqrt[n]{a}$ is a real number for all values of*a*.

- When
**Simplifying Odd and Even Roots**- For any integer $n\ge 2,$
- when
*n*is odd $\sqrt[n]{{a}^{n}}=a$ - when
*n*is even $\sqrt[n]{{a}^{n}}=\left|a\right|$

- when
- We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

- For any integer $n\ge 2,$

### 8.2 Simplify Radical Expressions

**Simplified Radical Expression**- For real numbers
*a*,*m*and $n\ge 2$

$\sqrt[n]{a}$ is considered simplified if*a*has no factors of ${m}^{n}$

- For real numbers
**Product Property of n**^{th}Roots- For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge 2$

$\sqrt[n]{ab}=\sqrt[n]{a}\xb7\sqrt[n]{b}$ and $\sqrt[n]{a}\xb7\sqrt[n]{b}=\sqrt[n]{ab}$

- For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge 2$
**How to simplify a radical expression using the Product Property**- Step 1. Find the largest factor in the radicand that is a perfect power of the index.

Rewrite the radicand as a product of two factors, using that factor. - Step 2. Use the product rule to rewrite the radical as the product of two radicals.
- Step 3. Simplify the root of the perfect power.

- Step 1. Find the largest factor in the radicand that is a perfect power of the index.
**Quotient Property of Radical Expressions**- If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne 0,$ and for any integer $n\ge 2$ then,

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

- If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne 0,$ and for any integer $n\ge 2$ then,
**How to simplify a radical expression using the Quotient Property.**- Step 1. Simplify the fraction in the radicand, if possible.
- Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
- Step 3. Simplify the radicals in the numerator and the denominator.

### 8.3 Simplify Rational Exponents

**Rational Exponent ${a}^{\frac{1}{n}}$**- If $\sqrt[n]{a}$ is a real number and $n\ge 2,$ then ${a}^{\frac{1}{n}}=\sqrt[n]{a}.$

**Rational Exponent ${a}^{\frac{m}{n}}$**- For any positive integers
*m*and*n*,

${a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}$ and ${a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}$

- For any positive integers
**Properties of Exponents**- If
*a, b*are real numbers and*m, n*are rational numbers, then**Product Property**${a}^{m}\xb7{a}^{n}={a}^{m+n}$**Power Property**${\left({a}^{m}\right)}^{n}={a}^{m\xb7n}$**Product to a Power**${\left(ab\right)}^{m}={a}^{m}{b}^{m}$**Quotient Property**$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}a\ne 0$**Zero Exponent Definition**${a}^{0}=1,$ $a\ne 0$**Quotient to a Power Property**${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.2em}{0ex}}b\ne 0$**Negative Exponent Property**${a}^{\text{\u2212}n}=\frac{1}{{a}^{n}},a\ne 0$

- If

### 8.4 Add, Subtract, and Multiply Radical Expressions

**Product Property of Roots**- For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge 2$

$\sqrt[n]{ab}=\sqrt[n]{a}\xb7\sqrt[n]{b}$ and $\sqrt[n]{a}\xb7\sqrt[n]{b}=\sqrt[n]{ab}$

- For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge 2$
**Special Products**

$\begin{array}{cccccc}\hfill \mathbf{\text{Binomial Squares}}\hfill & & & & & \hfill \mathbf{\text{Product of Conjugates}}\hfill \\ \hfill {\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}\hfill & & & & & \hfill \left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}\hfill \\ \hfill {\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}\hfill & & & & & \end{array}$

### 8.5 Divide Radical Expressions

**Quotient Property of Radical Expressions**- If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne 0,$ and for any integer $n\ge 2$ then,

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

- If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne 0,$ and for any integer $n\ge 2$ then,
**Simplified Radical Expressions**- A radical expression is considered simplified if there are:
- no factors in the radicand that have perfect powers of the index
- no fractions in the radicand
- no radicals in the denominator of a fraction

- A radical expression is considered simplified if there are:

### 8.6 Solve Radical Equations

**Binomial Squares**

$\begin{array}{c}{\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}\\ {\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}\end{array}$**Solve a Radical Equation**- Step 1. Isolate one of the radical terms on one side of the equation.
- Step 2. Raise both sides of the equation to the power of the index.
- Step 3. Are there any more radicals?

If yes, repeat Step 1 and Step 2 again.

If no, solve the new equation. - Step 4. Check the answer in the original equation.

**Problem Solving Strategy for Applications with Formulas**- Step 1. Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
- Step 2. Identify what we are looking for.
- Step 3. Name what we are looking for by choosing a variable to represent it.
- Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Step 5. Solve the equation using good algebra techniques.
- Step 6. Check the answer in the problem and make sure it makes sense.
- Step 7. Answer the question with a complete sentence.

**Falling Objects**- On Earth, if an object is dropped from a height of
*h*feet, the time in seconds it will take to reach the ground is found by using the formula $t=\frac{\sqrt{h}}{4}.$

- On Earth, if an object is dropped from a height of
**Skid Marks and Speed of a Car**- If the length of the skid marks is
*d*feet, then the speed,*s*, of the car before the brakes were applied can be found by using the formula $s=\sqrt{24d}.$

- If the length of the skid marks is

### 8.7 Use Radicals in Functions

**Properties of $\sqrt[n]{a}$**- When
*n*is an**even**number and:

$a\ge 0,$ then $\sqrt[n]{a}$ is a real number.

$a<0,$ then $\sqrt[n]{a}$ is not a real number. - When
*n*is an**odd**number, $\sqrt[n]{a}$ is a real number for all values of*a*.

- When
**Domain of a Radical Function**- When the
**index**of the radical is**even**, the radicand must be greater than or equal to zero. - When the
**index**of the radical is**odd**, the radicand can be any real number.

- When the

### 8.8 Use the Complex Number System

**Square Root of a Negative Number**- If
*b*is a positive real number, then $\sqrt{\text{\u2212}b}=\sqrt{b}i$

$a+bi$ $b=0$ $\begin{array}{}\\ a+0\xb7i\\ \\ \phantom{\rule{1.2em}{0ex}}a\end{array}$ Real number $b\ne 0$ $a+bi$ Imaginary number $a=0$ $\begin{array}{l}0+bi\\ \\ \\ \phantom{\rule{0.8em}{0ex}}bi\end{array}$ Pure imaginary number - A complex number is in
**standard form**when written as*a*+*bi*, where*a, b*are real numbers.

- If
**Product of Complex Conjugates**- If
*a, b*are real numbers, then

$\left(a-bi\right)\left(a+bi\right)={a}^{2}+{b}^{2}$

- If
**How to Divide Complex Numbers**- Step 1. Write both the numerator and denominator in standard form.
- Step 2. Multiply the numerator and denominator by the complex conjugate of the denominator.
- Step 3. Simplify and write the result in standard form.